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Effective equations governing an active poroelastic medium. (English) Zbl 1404.74040

Summary: In this work, we consider the spatial homogenization of a coupled transport and fluid-structure interaction model, to the end of deriving a system of effective equations describing the flow, elastic deformation and transport in an active poroelastic medium. The ‘active’ nature of the material results from a morphoelastic response to a chemical stimulant, in which the growth time scale is strongly separated from other elastic time scales. The resulting effective model is broadly relevant to the study of biological tissue growth, geophysical flows (e.g. swelling in coals and clays) and a wide range of industrial applications (e.g. absorbant hygiene products). The key contribution of this work is the derivation of a system of homogenized partial differential equations describing macroscale growth, coupled to transport of solute, that explicitly incorporates details of the structure and dynamics of the microscopic system, and, moreover, admits finite growth and deformation at the pore scale. The resulting macroscale model comprises a Biot-type system, augmented with additional terms pertaining to growth, coupled to an advection-reaction-diffusion equation. The resultant system of effective equations is then compared with other recent models under a selection of appropriate simplifying asymptotic limits.

MSC:

74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74L15 Biomechanical solid mechanics
92C10 Biomechanics
76V05 Reaction effects in flows
35Q35 PDEs in connection with fluid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
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